lennard_jones

All data and functions for computing the Lennard-Jones molecular diameter.

thermosteam.properties.lennard_jones.Stockmayer(Tm=None, Tb=None, Tc=None, Zc=None, omega=None, CASRN='', AvailableMethods=False, Method=None)[source]

This function handles the retrieval or calculation a chemical’s Stockmayer parameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 7 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available.

Prefered sources are ‘Magalhães, Lito, Da Silva, and Silva (2013)’ for common chemicals which had valies listed in that source, and the CSP method Tee, Gotoh, and Stewart CSP with Tc, omega (1966) for chemicals which don’t.

Examples

>>> Stockmayer(CASRN='64-17-5')
1291.41
Parameters
  • Tm (float, optional) – Melting temperature of fluid [K]

  • Tb (float, optional) – Boiling temperature of fluid [K]

  • Tc (float, optional) – Critical temperature, [K]

  • Zc (float, optional) – Critical compressibility, [-]

  • omega (float, optional) – Acentric factor of compound, [-]

  • CASRN (string, optional) – CASRN [-]

Returns

  • epsilon_k (float) – Lennard-Jones depth of potential-energy minimum over k, [K]

  • methods (list, only returned if AvailableMethods == True) – List of methods which can be used to obtain epsilon with the given inputs

Other Parameters
  • Method (string, optional) – A string for the method name to use, as defined by constants in Stockmayer_methods

  • AvailableMethods (bool, optional) – If True, function will determine which methods can be used to obtain epsilon for the desired chemical, and will return methods instead of epsilon

Notes

These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior.

The tabulated data is from [2]_, for 322 chemicals.

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

2

Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. “Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density.” The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.

thermosteam.properties.lennard_jones.molecular_diameter(Tc=None, Pc=None, Vc=None, Zc=None, omega=None, Vm=None, Vb=None, CASRN='', AvailableMethods=False, Method=None)[source]

This function handles the retrieval or calculation a chemical’s L-J molecular diameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 9 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available.

Prefered sources are ‘Magalhães, Lito, Da Silva, and Silva (2013)’ for common chemicals which had valies listed in that source, and the CSP method Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966) for chemicals which don’t.

Examples

>>> molecular_diameter(CASRN='64-17-5')
4.23738
Parameters
  • Tc (float, optional) – Critical temperature, [K]

  • Pc (float, optional) – Critical pressure, [Pa]

  • Vc (float, optional) – Critical volume, [m^3/mol]

  • Zc (float, optional) – Critical compressibility, [-]

  • omega (float, optional) – Acentric factor of compound, [-]

  • Vm (float, optional) – Molar volume of liquid at the melting point of the fluid [K]

  • Vb (float, optional) – Molar volume of liquid at the boiling point of the fluid [K]

  • CASRN (string, optional) – CASRN [-]

Returns

  • sigma (float) – Lennard-Jones molecular diameter, [Angstrom]

  • methods (list, only returned if AvailableMethods == True) – List of methods which can be used to obtain epsilon with the given inputs

Other Parameters
  • Method (string, optional) – A string for the method name to use, as defined by constants in molecular_diameter_methods

  • AvailableMethods (bool, optional) – If True, function will determine which methods can be used to obtain sigma for the desired chemical, and will return methods instead of sigma

Notes

These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior.

The tabulated data is from [2]_, for 322 chemicals.

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

2

Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. “Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density.” The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.

thermosteam.properties.lennard_jones.sigma_Flynn(Vc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [1]_ as reported in [2]_.

\[\sigma = 0.561(V_c^{1/3})^{5/4}\]
Parameters

Vc (float) – Critical volume of fluid [m^3/mol]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Vc is originally in units of mL/mol.

Examples

>>> sigma_Flynn(0.000268)
5.2506948422196285

References

1

Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).

2

Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

thermosteam.properties.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1]_.

\[\sigma = 2.44(T_c/P_c)^{1/3}\]
Parameters
  • Tc (float) – Critical temperature of fluid [K]

  • Pc (float) – Critical pressure of fluid [Pa]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Original units of critical pressure are atmospheres.

Examples

>>> sigma_Bird_Stewart_Lightfoot_critical_2(560.1, 4550000)
5.658657684653222

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

thermosteam.properties.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_1(Vc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [1]_.

\[\sigma = 0.841 V_c^{1/3}\]
Parameters

Vc (float) – Critical volume of fluid [m^3/mol]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Original units of Vc are mL/mol.

Examples

>>> sigma_Bird_Stewart_Lightfoot_critical_1(0.000268)
5.422184116631474

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

thermosteam.properties.lennard_jones.sigma_Bird_Stewart_Lightfoot_boiling(Vb)[source]

Calculates Lennard-Jones molecular diameter. Uses molar volume of liquid at boiling. CSP method by [1]_.

\[\sigma = 1.166V_{b,liq}^{1/3}\]
Parameters

Vb (float) – Boiling molar volume of liquid [m^3/mol]

Returns

sigma – Lennard-Jones collision integral, [Angstrom]

Return type

float

Notes

Original units of Vb are mL/mol.

Examples

>>> sigma_Bird_Stewart_Lightfoot_boiling(0.0001015)
5.439018856944655

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

thermosteam.properties.lennard_jones.sigma_Bird_Stewart_Lightfoot_melting(Vm)[source]

Calculates Lennard-Jones molecular diameter. Uses molar volume of a liquid at its melting point. CSP method by [1]_.

\[\sigma = 1.222 V_{m,sol}^{1/3}\]
Parameters

Vm (float) – Melting molar volume of a liquid at its melting point [m^3/mol]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Original units of Vm are mL/mol.

Examples

>>> sigma_Bird_Stewart_Lightfoot_melting(8.8e-05)
5.435407341351406

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

thermosteam.properties.lennard_jones.sigma_Stiel_Thodos(Vc, Zc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume and compressibility. CSP method by [1]_.

\[\sigma = 0.1866 V_c^{1/3} Z_c^{-6/5}\]
Parameters
  • Vc (float) – Critical volume of fluid [m^3/mol]

  • Zc (float) – Critical compressibility of fluid, [-]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Vc is originally in units of mL/mol.

Examples

Monofluorobenzene

>>> sigma_Stiel_Thodos(0.000271, 0.265)
5.94300853971033

References

1

Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

thermosteam.properties.lennard_jones.sigma_Tee_Gotoh_Steward_1(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1]_.

\[\sigma = 2.3647 \left(\frac{T_c}{P_c}\right)^{1/3}\]
Parameters
  • Tc (float) – Critical temperature of fluid [K]

  • Pc (float) – Critical pressure of fluid [Pa]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [1]_ but are not included here, except for sigma_Tee_Gotoh_Steward_2.

Examples

>>> sigma_Tee_Gotoh_Steward_1(560.1, 4550000)
5.48402779790962

References

1

Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

thermosteam.properties.lennard_jones.sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature, pressure, and acentric factor. CSP method by [1]_.

\[\sigma = (2.3551 - 0.0874\omega)\left(\frac{T_c}{P_c}\right)^{1/3}\]
Parameters
  • Tc (float) – Critical temperature of fluid [K]

  • Pc (float) – Critical pressure of fluid [Pa]

  • omega (float) – Acentric factor for fluid, [-]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [1]_ but are not included here, except for sigma_Tee_Gotoh_Steward_1.

Examples

>>> sigma_Tee_Gotoh_Steward_2(560.1, 4550000, 0.245)
5.412104867264477

References

1

Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

thermosteam.properties.lennard_jones.sigma_Silva_Liu_Macedo(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1]_.

\[\sigma_{LJ}^3 = 0.17791 + 11.779 \left( \frac{T_c}{P_c}\right) - 0.049029\left( \frac{T_c}{P_c}\right)^2\]
Parameters
  • Tc (float) – Critical temperature of fluid [K]

  • Pc (float) – Critical pressure of fluid [Pa]

Returns

sigma – Lennard-Jones molecular diameter, [Angstrom]

Return type

float

Notes

Pc is originally in bar. An excellent paper. None is returned if the polynomial returns a negative number, as in the case of 1029.13 K and 3.83 bar.

Examples

>>> sigma_Silva_Liu_Macedo(560.1, 4550000)
5.164483998730177

References

1

Silva, Carlos M., Hongqin Liu, and Eugenia A. Macedo. “Models for Self-Diffusion Coefficients of Dense Fluids, Including Hydrogen-Bonding Substances.” Chemical Engineering Science 53, no. 13 (July 1, 1998): 2423-29. doi:10.1016/S0009-2509(98)00037-2

thermosteam.properties.lennard_jones.epsilon_Flynn(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [1]_ as reported in [2]_.

\[\epsilon/k = 1.77 T_c^{5/6}\]
Parameters

Tc (float) – Critical temperature of fluid [K]

Returns

epsilon_k – Lennard-Jones depth of potential-energy minimum over k, [K]

Return type

float

Notes

Examples

>>> epsilon_Flynn(560.1)
345.2984087011443

References

1

Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).

2

Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

thermosteam.properties.lennard_jones.epsilon_Bird_Stewart_Lightfoot_critical(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [1]_.

\[\epsilon/k = 0.77T_c\]
Parameters

Tc (float) – Critical temperature of fluid [K]

Returns

epsilon_k – Lennard-Jones depth of potential-energy minimum over k, [K]

Return type

float

Notes

Examples

>>> epsilon_Bird_Stewart_Lightfoot_critical(560.1)
431.27700000000004

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

thermosteam.properties.lennard_jones.epsilon_Bird_Stewart_Lightfoot_boiling(Tb)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses boiling temperature. CSP method by [1]_.

\[\epsilon/k = 1.15 T_b\]
Parameters

Tb (float) – Boiling temperature [K]

Returns

epsilon_k – Lennard-Jones depth of potential-energy minimum over k, [K]

Return type

float

Notes

Examples

>>> epsilon_Bird_Stewart_Lightfoot_boiling(357.85)
411.5275

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

thermosteam.properties.lennard_jones.epsilon_Bird_Stewart_Lightfoot_melting(Tm)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses melting temperature. CSP method by [1]_.

\[\epsilon/k = 1.92T_m\]
Parameters

Tm (float) – Melting temperature [K]

Returns

epsilon_k – Lennard-Jones depth of potential-energy minimum over k, [K]

Return type

float

Notes

Examples

>>> epsilon_Bird_Stewart_Lightfoot_melting(231.15)
443.808

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

thermosteam.properties.lennard_jones.epsilon_Stiel_Thodos(Tc, Zc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature and critical compressibility. CSP method by [1]_.

\[\epsilon/k = 65.3 T_c Z_c^{3.6}\]
Parameters
  • Tc (float) – Critical temperature of fluid [K]

  • Zc (float) – Critical compressibility of fluid, [-]

Returns

epsilon_k – Lennard-Jones depth of potential-energy minimum over k, [K]

Return type

float

Notes

Examples

Fluorobenzene

>>> epsilon_Stiel_Thodos(358.5, 0.265)
196.3755830305783

References

1

Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

thermosteam.properties.lennard_jones.epsilon_Tee_Gotoh_Steward_1(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature. CSP method by [1]_.

\[\epsilon/k = 0.7740T_c\]
Parameters

Tc (float) – Critical temperature of fluid [K]

Returns

epsilon_k – Lennard-Jones depth of potential-energy minimum over k, [K]

Return type

float

Notes

Further regressions with other parameters were performed in [1]_ but are not included here, except for epsilon_Tee_Gotoh_Steward_2.

Examples

>>> epsilon_Tee_Gotoh_Steward_1(560.1)
433.5174

References

1

Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

thermosteam.properties.lennard_jones.epsilon_Tee_Gotoh_Steward_2(Tc, omega)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature and acentric factor. CSP method by [1]_.

\[\epsilon/k = (0.7915 + 0.1693 \omega)T_c\]
Parameters
  • Tc (float) – Critical temperature of fluid [K]

  • omega (float) – Acentric factor for fluid, [-]

Returns

epsilon_k – Lennard-Jones depth of potential-energy minimum over k, [K]

Return type

float

Notes

Further regressions with other parameters were performed in [1]_ but are not included here, except for epsilon_Tee_Gotoh_Steward_1.

Examples

>>> epsilon_Tee_Gotoh_Steward_2(560.1, 0.245)
466.55125785

References

1

Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

thermosteam.properties.lennard_jones.collision_integral_Neufeld_Janzen_Aziz(Tstar, l=1, s=1)[source]

Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 100. Values are accurate to 0.1 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [1]_.

\[\Omega_D = \frac{A}{T^{*B}} + \frac{C}{\exp(DT^*)} + \frac{E}{\exp(FT^{*})} + \frac{G}{\exp(HT^*)} + RT^{*B}\sin(ST^{*W}-P)\]
Parameters
  • Tstar (float) – Reduced temperature of the fluid [-]

  • l (int) – term

  • s (int) – term

Returns

Omega – Collision integral of A and B

Return type

float

Notes

Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4).

\[T^* = \frac{k_b T}{\epsilon}\]

Results are very similar to those of the more modern formulation, collision_integral_Kim_Monroe.

Calculations begin to yield overflow errors in some values of (l, 2) after Tstar = 75, beginning with (1, 7). Also susceptible are (1, 5) and (1, 6).

Examples

>>> collision_integral_Neufeld_Janzen_Aziz(100, 1, 1)
0.516717697672334

References

1

Neufeld, Philip D., A. R. Janzen, and R. A. Aziz. “Empirical Equations to Calculate 16 of the Transport Collision Integrals Omega(l, S)* for the Lennard-Jones (12-6) Potential.” The Journal of Chemical Physics 57, no. 3 (August 1, 1972): 1100-1102. doi:10.1063/1.1678363

thermosteam.properties.lennard_jones.collision_integral_Kim_Monroe(Tstar, l=1, s=1)[source]

Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 400. Values are accurate to 0.007 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [1]_.

\[\Omega^{(l,s)*} = A^{(l,s)} + \sum_{k=1}^6 \left[ \frac{B_k^{(l,s)}} {(T^*)^k} + C_k^{(l,s)} (\ln T^*)^k \right]\]
Parameters
  • Tstar (float) – Reduced temperature of the fluid [-]

  • l (int) – term

  • s (int) – term

Returns

Omega – Collision integral of A and B

Return type

float

Notes

Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4).

\[T^* = \frac{k_b T}{\epsilon}\]

Examples

>>> collision_integral_Kim_Monroe(400, 1, 1)
0.4141818082392228

References

1

Kim, Sun Ung, and Charles W. Monroe. “High-Accuracy Calculations of Sixteen Collision Integrals for Lennard-Jones (12-6) Gases and Their Interpolation to Parameterize Neon, Argon, and Krypton.” Journal of Computational Physics 273 (September 15, 2014): 358-73. doi:10.1016/j.jcp.2014.05.018.

thermosteam.properties.lennard_jones.Tstar(T, epsilon_k=None, epsilon=None)[source]

This function calculates the parameter Tstar as needed in performing collision integral calculations.

\[T^* = \frac{kT}{\epsilon}\]

Examples

>>> Tstar(T=318.2, epsilon_k=308.43)
1.0316765554582887
Parameters
  • epsilon_k (float, optional) – Lennard-Jones depth of potential-energy minimum over k, [K]

  • epsilon (float, optional) – Lennard-Jones depth of potential-energy minimum [J]

Returns

Tstar – Dimentionless temperature for calculating collision integral, [-]

Return type

float

Notes

Tabulated values are normally listed as epsilon/k. k is the Boltzman constant, with units of J/K.

References

1

Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006